Yet the later the hour grows for the bill’s approval by the two governors, the more skittish its sponsors are growing about its chances of becoming law.

Gov. Andrew M. Cuomo faces the first deadline: He has until Saturday night at midnight  to decide on the New York bill. If he approves it, Gov. Chris Christieof New Jersey will face heightened pressure to do the same; his deadline is in mid-January. But because the authority is a creation of the two states, approval of the legislation by only one state would have the same effect as its rejection by both.

Supporters of the bill say they now fear that the governors will reject the bill because it would erode the control they share evenly over everything the Port Authority does — whether running airports and bridges, hiring politicallyconnected officials or awarding billion-dollar contracts to benefit political allies.

Neither Mr. Christie, a Republican, nor Mr. Cuomo, a Democrat, has signaled his intention. The two had lunch together in Carlstadt, N.J., on Tuesday, as NJ.com reported, but spokesmen for both declined to say whether the two had discussed the Port Authority bill, let alone whether they planned to approve it.

The impact of the legislation would be profound, sponsors say: It imposes a broad array of new financial, ethical and administrative controls on the Port Authority, including a requirement that all its meetings be open to the public. It also requires its commissioners to acknowledge that they have a “fiduciary duty” to the Port Authority, a “duty of loyalty and care” to the authority, and an obligation to act in its best interests and that of the people it serves.

In other words, said Richard L. Brodsky, a former Democratic assemblyman from Westchester County who has long been a critic of the Port Authority’s governance: “It requires board members to promise loyalty to the authority. Not to the guys who appointed them.”

The stakes in Trenton are somewhat higher than in Albany. Mr. Christie and the Port Authority, after all, were central characters in the lane-closing scandal at the George Washington Bridge in September 2013. Federal and Manhattan investigators are still looking into that event and related questions about Port Authority finances that emerged later. Mr. Christie has denied knowing anything about the lane closings in advance. But the scrutiny damaged his national image as he planned a potential 2016 presidential run, and his stewardship of the Port Authority remains a political problem that could flare up again at any time.

With that in mind, backers of the legislation say they fear a pre-emptive strike on the New York side of the river.

In early May, Mr. Cuomo and Mr. Christie formed a special panel on the Port Authority’s future. Were that panel to release findings before Saturday, the theory goes, it could give Mr. Cuomo a rationale for vetoing the legislation, which in turn would spare Mr. Christie the decision.

“The governors’ special panel should not cancel out the good work done so far with these unanimously supported bistate bills,” said Dick Dadey, executive director of Citizens Union, a government watchdog group in New York. He added, “These unanimously supported reform bills should be embraced by these two reform-minded governors.”

Supporters in both states are bracing for disappointment while simultaneously expressing disbelief at the prospect.

“It’s not just bipartisan and bistate: it’s 612 to zero,” State Senator Robert M. Gordon, a Democrat of Bergen County, N.J., said of his count of the unanimous two-state legislative vote. “How could they not sign it?”

That may become clear presently.

Proponents of the bill in Albany met with Mr. Cuomo’s legal staff and wrote to the governor in mid-December. But they said they had received no indication of Mr. Cuomo’s plans.

“They have not been specific about any objections,” said Assemblyman James F. Brennan, a Brooklyn Democrat who is the bill’s chief sponsor in that chamber. Mr. Brennan said there was “no possible merit to a veto,” noting that many of the reforms proposed for the Port Authority were already in place in other state authorities and agencies. “This is standard modern ethics reform,” he added.

But as the clock ticks, backers of the measure say they fear that if the bills fail, it will be difficult to muster such support again.

“It would be very challenging to get both states and both sides of the aisle to agree on every point,” said Assemblywoman Valerie Vainieri Huttle, a Democrat of Bergen County, N.J.

The lobbying has continued. Mr. Gordon said he pressed the case with Governor Christie at a recent holiday party. “I literally accosted him,” Mr. Gordon said.

It was an unsuccessful ambush. “I didn’t expect to get a clear answer,” he said. “And I didn’t.”